Universal Polynomial Research Articles

Universal polynomials for Severi degrees of toric surfaces

The Severi variety parameterizes plane curves of degree d with δ nodes. Its degree is called the Severi degree. For large enough d, the Severi degrees coincide with the Gromov–Witten invariants of CP2. Fomin and Mikhalkin (2010) [10] proved the 1995 conjecture that for fixed δ, Severi degrees are eventually polynomial in d.In this paper, we study the Severi varieties corresponding to a large family of toric surfaces. We prove the analogous result that the Severi degrees are eventually polynomial as a function of the multidegree. More surprisingly, we show that the Severi degrees are also eventually polynomial “as a function of the surface”. We illustrate our theorems by explicitly computing, for a small number of nodes, the Severi degree of any large enough Hirzebruch surface and of a singular surface.Our strategy is to use tropical geometry to express Severi degrees in terms of Brugallé and Mikhalkin’s floor diagrams, and study those combinatorial objects in detail. An important ingredient in the proof is the polynomiality of the discrete volume of a variable facet-unimodular polytope.

Twisted Alexander Polynomials of Hyperbolic Knots

We study a twisted Alexander polynomial naturally associated to a hyperbolic knot in an integer homology 3-sphere via a lift of the holonomy representation to . It is an unambiguous symmetric Laurent polynomial whose coefficients lie in the trace field of the knot. It contains information about genus, fibering, and chirality, and moreover, is powerful enough to sometimes detect mutation. We calculated this invariant numerically for all 313 209 hyperbolic knots in S 3 with at most 15 crossings, and found that in all cases it gave a sharp bound on the genus of the knot and determined both fibering and chirality. We also study how such twisted Alexander polynomials vary as one moves around in an irreducible component X 0 of the -character variety of the knot group. We show how to understand all of these polynomials at once in terms of a polynomial whose coefficients lie in the function field of X 0. We use this to help explain some of the patterns observed for knots in S 3, and explore a potential relationship between this universal polynomial and the Culler–Shalen theory of surfaces associated to ideal points.

Action du groupe des opérateurs de gamétisation sur les identités ω-polynomiales

The gametization process reduces the study of non-commutative and non-associative algebras satisfying non-homogeneous polynomial identities with variables in X={x1,…,xn} to algebras verifying simpler identities. However after a gametization, certain identities remain invariant and other identities, said universal invariant, are invariant for every gametization. Now in the case n=1, for all algebras satisfying a universal invariant polynomial identity studied until now, we know that the existence of an idempotent is not certain. Using an action of the gametization operators group on the non-commutative and non-associative algebra of polynomials K〈X〉, we give all identities which are invariant and universal invariant by gametization.

Universal Polynomial Expansions of Harmonic Functions

Let Ω be a domain in ℝN such that \(\left(\mathbb{R}^{N}\cup\lbrace\infty\rbrace\right)\setminus\Omega\) is connected. It is known that, for each w ∈ Ω, there exist harmonic functions on Ω that are universal at w, in the sense that partial sums of their homogeneous polynomial expansion about w approximate all plausibly approximable functions in the complement of Ω. Under the assumption that Ω omits an infinite cone, it is shown that the connectedness hypothesis on \(\left(\mathbb{R}^{N}\cup\lbrace\infty\rbrace\right)\setminus\Omega\) is essential, and that a harmonic function which is universal at one point is actually universal at all points of Ω.

A proof of the Göttsche-Yau-Zaslow formula

Let $S$ be a complex smooth projective surface and $L$ be a line bundle on $S$. Göttsche conjectured that for every integer $r$, the number of $r$-nodal curves in $\left| L\right|$ is a universal polynomial of four topological numbers when $L$ is sufficiently ample. We prove Göttsche’s conjecture using the algebraic cobordism group of line bundles on surfaces and degeneration of Hilbert schemes of points. In addition, we prove the Göttsche-Yau-Zaslow Formula which expresses the generating function of the numbers of nodal curves in terms of quasimodular forms and two unknown series.

PARITY OF THE PARTITION FUNCTION AND TRACES OF SINGULAR MODULI

We prove that the parity of the partition function is given by the "trace" of the Hauptmodul [Formula: see text] for [Formula: see text] at points of complex multiplication. Using Hecke operators, we generalize this to relate the Hecke traces of [Formula: see text] to the partition function modulo 2. We then prove that the generating function for these Hecke traces is equal to the logarithmic derivative of the level 6 Hilbert class polynomial. Finally, we give a procedure involving Hilbert class polynomials for computing the parity of the partition function, and make some speculations about the distribution of these universal polynomials modulo class polynomials.

Asymptotics of characters of symmetric groups: Structure of Kerov character polynomials

We study asymptotics of characters of the symmetric groups on a fixed conjugacy class. It was proved by Kerov that such a character can be expressed as a polynomial in free cumulants of the Young diagram (certain functionals describing the shape of the Young diagram). We show that for each genus there exists a universal symmetric polynomial which gives the coefficients of the part of Kerov character polynomials with the prescribed homogeneous degree. The existence of such symmetric polynomials was conjectured by Lassalle.

Universal Polynomials for Severi Degrees of Toric Surfaces

The Severi variety parametrizes plane curves of degree $d$ with $\delta$ nodes. Its degree is called the Severi degree. For large enough $d$, the Severi degrees coincide with the Gromov-Witten invariants of $\mathbb{CP}^2$. Fomin and Mikhalkin (2009) proved the 1995 conjecture that for fixed $\delta$, Severi degrees are eventually polynomial in $d$. In this paper, we study the Severi varieties corresponding to a large family of toric surfaces. We prove the analogous result that the Severi degrees are eventually polynomial as a function of the multidegree. More surprisingly, we show that the Severi degrees are also eventually polynomial "as a function of the surface". Our strategy is to use tropical geometry to express Severi degrees in terms of Brugallé and Mikhalkin's floor diagrams, and study those combinatorial objects in detail. An important ingredient in the proof is the polynomiality of the discrete volume of a variable facet-unimodular polytope. La variété de Severi paramétrise les courbes planes de degré $d$ avec $\delta$ nœuds. Son degré s'appelle le degré de Severi. Pour $d$ assez grand, les degrés de Severi coïncident avec les invariants de Gromov-Witten de $\mathbb{CP}^2$. Fomin et Mikhalkin (2009) ont prouvé une conjecture de 1995 que pour $\delta$ fixé, les degrés de Severi sont à terme des polynômes en $d$. Nous étudions les variétés de Severi correspondant à une large famille de surfaces toriques. Nous prouvons le résultat analogue que les degrés de Severi sont à terme des fonctions polynomiales du multidegré. De manière plus surprenante, nous montrons que les degrés de Severi sont à terme des polynômes en tant que "fonction de la surface''. Notre stratégie est d'utiliser la géométrie tropicale pour exprimer les degrés de Severi en fonction des "floor diagrams" de Brugallé et Mikhalkin, et d'utiliser ces objets combinatoires en détail. Un autre ingrédient important de la preuve est la polynomialité du volume discret d'un polytope face-unimodulaire variable.

Estimating topology of complex networks based on sparse Bayesian learning

We propose a method of estimating complex network topology with a noisy environment. Our method can estimate not only dynamical equation of the chaotic system and its parameters but also topology, the dynamical equation of each node, all the parameters, coupling direction and coupling strength of complex dynamical network composed of coupled unknown chaotic systems using only noisy time series. Estimating the system structure and parameter is regard as estimating the linear regression coefficients by reconstructing system with universal polynomial structure. Reconstruction algorithm of Bayesian compressive sensing is used for estimating the coefficients of regression polynomial. For the reconstruction from noisy time series we adopt relevance vector machine, namely we use sparse Bayesian learning to solve sparse undetermined linear equation to obtain the objects mentioned above. The Lorenz system and a scale free network composed of 200 Lorenz systems are provided to illustrate the efficiency. Simulation results show that our method improves the robust to noise compared with the compressive sensing and has fast convergence speed and tiny steady state error compared with the least square strategy.

The Universal Edge Elimination Polynomial and the Dichromatic Polynomial

The dichromatic polynomial Z(G;q,v) can be characterized as the most general C-invariant, i.e., a graph polynomial satisfying a linear recurrence with respect to edge deletion and edge contraction. Similarly, the universal edge elimination polynomial ξ(G;x,y,z) introduced in [Ilya Averbouch, Benny Godlin, and Johann A. Makowsky. An extension of the bivariate chromatic polynomial. Eur. J. Comb, 31(1):1–17, 2010] can be characterized as the most general EE-invariant, i.e., a graph polynomial satisfying a linear recurrence with respect to edge deletion, edge contraction and edge extraction. In this paper we examine substitution instances of ξ(G;x,y,z) and show that among these the dichromatic polynomial Z(G;q,v) plays a distinctive role.

Eighth-order high-temperature expansion for general Heisenberg Hamiltonians

We explicitly calculate the moments $t_n$ of general Heisenberg Hamiltonians up to eighth order. They have the form of finite sums of products of two factors. The first factor is represented by a (multi-)graph which has to be evaluated for each particular system under consideration. The second factors are well-known universal polynomials in the variable $s(s+1)$, where $s$ denotes the individual spin quantum number. From these moments we determine the corresponding coefficients of the high-temperature expansion of the free energy and the zero field susceptibility by a new method. These coefficients can be written in a form which makes explicit their extensive character. Our results represent a general tool to calculate eighth-order high-temperature series for arbitrary Heisenberg models. The results are applied to concrete systems, namely to magnetic molecules with the geometry of the icosidodecahedron, to frustrated square lattices, and to the pyrochlore magnets. By comparison with other methods that have been recently applied to these systems, we find that the typical susceptibility maximum of the spin-$s$ Heisenberg antiferromagnet is well described by the eighth-order high-temperature series.

The enumeration of vertex induced subgraphs with respect to the number of components

Inspired by the study of community structure in connection networks, we introduce the graph polynomial Q ( G ; x , y ) , the bivariate generating function which counts the number of connected components in induced subgraphs. We give a recursive definition of Q ( G ; x , y ) using vertex deletion, vertex contraction and deletion of a vertex together with its neighborhood and prove a universality property. We relate Q ( G ; x , y ) to other known graph invariants and graph polynomials, among them partition functions, the Tutte polynomial, the independence and matching polynomials, and the universal edge elimination polynomial introduced by I. Averbouch et al. (2008) [5]. We show that Q ( G ; x , y ) is vertex reconstructible in the sense of Kelly and Ulam, and discuss its use in computing residual connectedness reliability. Finally we show that the computation of Q ( G ; x , y ) is ♯ P -hard, but fixed parameter tractable for graphs of bounded tree-width and clique-width.

A short proof of the Göttsche conjecture

We prove that for a sufficiently ample line bundle $L$ on a surface $S$, the number of $\delta$-nodal curves in a general $\delta$-dimensional linear system is given by a universal polynomial of degree $\delta$ in the four numbers $L^2,\,L.K_S,\,K_S^2$ and $c_2(S)$. The technique is a study of Hilbert schemes of points on curves on a surface, using the BPS calculus of [PT3] and the computation of tautological integrals on Hilbert schemes by Ellingsrud, G\"ottsche and Lehn. We are also able to weaken the ampleness required, from G\"ottsche's $(5\delta-1)$-very ample to $\delta$-very ample.

Reconfigurable Hardware Implementations of Tweakable Enciphering Schemes

Tweakable enciphering schemes are length-preserving block cipher modes of operation that provide a strong pseudorandom permutation. It has been suggested that these schemes can be used as the main building blocks for achieving in-place disk encryption. In the past few years, there has been an intense research activity toward constructing secure and efficient tweakable enciphering schemes. But actual experimental performance data of these newly proposed schemes are yet to be reported. In this paper, we present optimized FPGA implementations of six tweakable enciphering schemes, namely, HCH, HCTR, XCB, EME, HEH, and TET, using a 128-bit AES core as the underlying block cipher. We report the performance timings of these modes when using both pipelined and sequential AES structures. The universal polynomial hash function included in the specification of HCH, HCHfp (a variant of HCH), HCTR, XCB, TET, and HEH was implemented using a Karatsuba multiplier as the main building block. We provide detailed algorithm analysis of each of the schemes trying to exploit their inherent parallelism as much as possible. Our experiments show that a sequential AES core is not an attractive option for the design of these modes as it leads to rather poor throughput. In contrast, according to our place-and-route results on a Xilinx Virtex 4 FPGA, our designs achieve a throughput of 3.95 Gbps for HEH when using an encryption/decryption pipelined AES core, and a throughput of 5.71 Gbps for EME when using a encryption-only pipeline AES core. The performance results reported in this paper provide experimental evidence that hardware implementations of tweakable enciphering schemes can actually match and even outperform the data rates achieved by state-of-the-art disk controllers, thus showing that they might be used for achieving provably secure in-place hard disk encryption.

Asymptotic fields ahead of mixed mode frictional cohesive cracks

AbstractThis paper discusses the asymptotic fields ahead of mixed mode frictional cohesive cracks in quasi‐brittle materials. These fields have been derived after reformatting the cohesive‐law in a special but universal polynomial containing fractional or integer powers. This special form ensures that the radial and angular variations of the asymptotic fields are separable as in the Williams expansions for a traction‐free crack. The coefficients of the expansions however depend nonlinearly on the softening law and the boundary conditions. As expected, the asymptotic field of a frictional cohesive crack reduces to that of a frictionless cohesive crack with normal cohesive separation when the friction coefficient becomes zero. It is also shown that the asymptotic field of a frictionless cohesive crack can be used to a non‐cohesive crack opened by crack face loading, after including the singular terms corresponding to the eigenvalue 1/2. Furthermore, the coefficients are given explicitly for two special softening laws that are commonly used in practice. The leading terms of the true displacement asymptotic field are also derived explicitly. These are especially useful as the enrichment functions at the tip of a mixed mode cohesive crack for accurate simulation of crack growth in quasi‐brittle materials using the extended finite element (XFEM).

Characteristic classes of [formula omitted]-manifolds: Classification and applications

A Q -manifold M is a supermanifold endowed with an odd vector field Q squaring to zero. The Lie derivative L Q along Q makes the algebra of smooth tensor fields on M into a differential algebra. In this paper, we define and study the invariants of Q -manifolds called characteristic classes. These take values in the cohomology of the operator L Q and, given an affine symmetric connection with curvature R , can be represented by universal tensor polynomials in the repeated covariant derivatives of Q and R up to some finite order. As usual, the characteristic classes are proved to be independent of the choice of the affine connection used to define them. The main result of the paper is a complete classification of the intrinsic characteristic classes, which, by definition, do not vanish identically on flat Q -manifolds. As an illustration of the general theory we interpret some of the intrinsic characteristic classes as anomalies in the BV and BFV-BRST quantization methods of gauge theories. An application to the theory of (singular) foliations is also discussed.

Topological graph polynomials and quantum field theory Part I: heat kernel theories

We investigate the relationship between the universal topological polynomials for graphs in mathematics and the parametric representation of Feynman amplitudes in quantum field theory. In this first article we consider translation invariant theories with the usual heat-kernel-based propagator. We show how the Symanzik polynomials of quantum field theory are particular multivariate versions of the Tutte polynomial, and how the new polynomials of noncommutative quantum field theory are special versions of the Bollobás–Riordan polynomials.

Stickelberger elements over rational function fields

The family of Stickelberger elements associated to certain abelian extensions over global rational function fields is represented by a universal polynomial. This result is then used to prove a conjecture of Gross for these Stickelberger elements.

On the reconstruction of graph invariants

The reconstruction conjecture has remained open for simple undirected graphs since it was suggested in 1941 by Kelly and Ulam. In an attempt to prove the conjecture, many graph invariants have been shown to be reconstructible from the vertex-deleted deck, and in particular, some prominent graph polynomials. Among these are the Tutte polynomial, the chromatic polynomial and the characteristic polynomial. We show that the interlace polynomial, the U -polynomial, the universal edge elimination polynomial xi and the colored versions of the latter two are reconstructible.

An extension of the bivariate chromatic polynomial

K. Dohmen, A. Pönitz and P. Tittmann [K. Dohmen, A. Pönitz, P. Tittmann, A new two-variable generalization of the chromatic polynomial, Discrete Mathematics and Theoretical Computer Science 6 (2003), 69–90], introduced a bivariate generalization of the chromatic polynomial P ( G , x , y ) which subsumes also the independent set polynomial of I. Gutman and F. Harary [I. Gutman, F. Harary, Generalizations of the matching polynomial, Utilitas Mathematicae 24 (1983), 97–106] and the vertex-cover polynomial of F.M. Dong, M.D. Hendy, K.T. Teo and C.H.C. Little [F.M. Dong, M.D. Hendy, K.L. Teo, and C.H.C. Little, The vertex-cover polynomial of a graph, Discrete Mathematics 250 (2002), 71–78]. We first show that P ( G , x , y ) has a recursive definition with respect to three kinds of edge eliminations: edge deletion, edge contraction, and edge extraction, i.e. deletion of an edge together with its endpoints. Like in the case of deletion and contraction only [J.G. Oxley and D.J.A. Welsh, The Tutte polynomial and percolation, in: J.A. Bundy, U.S.R. Murty (Eds.), Graph Theory and Related Topics, Academic Press, London, 1979, pp. 329–339] it turns out that there is a most general, or as they call it, a universal polynomial satisfying such recurrence relations with respect to the three kinds of edge eliminations, which we call ξ ( G , x , y , z ) . We show that the new polynomial simultaneously generalizes, P ( G , x , y ) , as well as the Tutte polynomial and the matching polynomial, We also give an explicit definition of ξ ( G , x , y , z ) using a subset expansion formula. We also show that ξ ( G , x , y , z ) can be viewed as a partition function, using counting of weighted graph homomorphisms. Furthermore, we expand this result to edge-labeled graphs as was done for the Tutte polynomial by T. Zaslavsky [T. Zaslavsky, Strong Tutte functions of matroids and graphs, Trans. Amer. Math. Soc. 334 (1992), 317–347] and by B. Bollobás and O. Riordan [B. Bollobás, O. Riordan, A Tutte polynomial for coloured graphs, Combinatorics, Probability and Computing 8 (1999), 45–94]. The edge-labeled polynomial ξ lab ( G , x , y , z , t ̄ ) also generalizes the chain polynomial of R.C. Read and E.G. Whitehead Jr. [R.C. Read, E.G. Whitehead Jr., Chromatic polynomials of homeomorphism classes of graphs, Discrete Mathematics 204 (1999), 337–356]. Finally, we discuss the complexity of computing ξ ( G , x , y , z ) .